COMPOSITION OPERATORS ON UNIFORM ALGEBRAS AND THE PSEUDOHYPERBOLIC METRIC

Title & Authors
COMPOSITION OPERATORS ON UNIFORM ALGEBRAS AND THE PSEUDOHYPERBOLIC METRIC
Galindo, P.; Gamelin, T.W.; Lindstrom, M.;

Abstract
Let A be a uniform algebra, and let $\small{\phi}$ be a self-map of the spectrum $\small{M_A}$ of A that induces a composition operator $\small{C_{\phi}}$, on A. It is shown that the image of $\small{M_A}$ under some iterate $\small{{\phi}^n}$ of \phi is hyperbolically bounded if and only if \phi has a finite number of attracting cycles to which the iterates of $\small{\phi}$ converge. On the other hand, the image of the spectrum of A under $\small{\phi}$ is not hyperbolically bounded if and only if there is a subspace of $\small{A^{**}}$ "almost" isometric to $\small{{\ell}_{\infty}}$ on which $\small{{C_{\phi}}^{**}}$ "almost" an isometry. A corollary of these characterizations is that if $\small{C_{\phi}}$ is weakly compact, and if the spectrum of A is connected, then $\small{\phi}$ has a unique fixed point, to which the iterates of $\small{\phi}$ converge. The corresponding theorem for compact composition operators was proved in 1980 by H. Kamowitz [17].
Keywords
uniform algebra;composition operator;hyperbolically bounded;interpolating sequence;
Language
English
Cited by
1.
Fredholm composition operators on algebras of analytic functions on Banach spaces, Journal of Functional Analysis, 2010, 258, 5, 1504
2.
Interpolating sequences on uniform algebras, Topology, 2009, 48, 2-4, 111
3.
Quasicompact and Riesz endomorphisms of Banach algebras, Journal of Functional Analysis, 2005, 225, 2, 427
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